Skip to main content
SpringerLink
Log in
Book cover

International Tbilisi Symposium on Logic, Language, and Computation

TbiLLC 2015: Logic, Language, and Computation pp 333–354Cite as

Subintuitionistic Logics with Kripke Semantics

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10148))

Abstract

The subintuitionistic logics introduced by Corsi and Restall are developed in a uniform manner. In this way Restall’s contributions are clarified. Hilbert type proof systems are given for derivations without and with assumptions. The results are applied to give conservation theorems for intuitionistic logic IPC over Corsi’s system F. For Visser’s basic logic additional conservation results are obtained.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    Neither is it the case that the class of NNIL-formulas is locally finite in BPC as it is in IPC, the fact that \(\nvdash _\mathsf{BPC} (p\rightarrow (p\rightarrow q))\leftrightarrow (p\rightarrow q)\) quickly leads to infinitely many non-equivalent NNIL-formulas in two variables.

References

  1. Aghaei, M., Ardeshir, M.: A bounded translation of intuitionistic propositional logic into basic propositional logic. Math. Logic Q. 46(2), 195–206 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Celani, S., Jansana, R.: A closer look at some subintuitionistic logics. Notre Dame J. Formal Logic. 42(4), 225–255 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Corsi, G.: Weak Logics with strict implication. Z. Math. Logik Grundlagen Math. 33(5), 389–406 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  4. Došen, K.: Modal translations in K and D. In: de Rijke, M. (ed.) Diamonds and Defaults. Synthese Library, vol. 229, pp. 103–127. Springer, Dordrecht (1994)

    Google Scholar 

  5. Kleene, S.C.: Disjunction and Existence under Implication in elementary Intuitionistic formalisms. J. Symbolic Logic 27, 11–18 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  6. Restall, G.: Subintuitionistic logics. Notre Dame J. Formal Logic 35(1), 116–129 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  7. Sano, K., Ma, M.: Alternative semantics for visser’s propositional logics. In: Aher, M., Hole, D., Jeřábek, E., Kupke, C. (eds.) TbiLLC 2013. LNCS, vol. 8984, pp. 257–275. Springer, Heidelberg (2015). doi:10.1007/978-3-662-46906-4_15

    Google Scholar 

  8. Shirmohammadzadeh Maleki, F., de Jongh, D.: Weak Subintuitionistic Logics. Logic J. IGPL (2016). doi:10.1093/jigpal/jzw062

  9. Suzuki, Y., Ono, H.: Hilbert-style proof system for BPL. Technical report, IS-RR-97-0040F. Japan Advanced Institute of Science and Technology (1997)

    Google Scholar 

  10. Visser, A.: A propositional logic with explicit fixed points. Studia Logica. 40(2), 155–175 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  11. Visser, A., van Benthem, J., de Jongh, D., Renardel de Lavalette, G.: NNIL, a study in intuitionistic propositional Logic. In: Ponse, A., de Rijke, M., Venema, Y. (eds.) Modal Logics and Process Algebra, a Bisimulation Perspective, pp. 289–326 (1995)

    Google Scholar 

Download references

Acknowledgement

We thank M. Ardeshir for important comments on the paper. We are grateful to two unknown referees who provided valuable remarks and corrections.

Author information

Authors and Affiliations

  1. Institute for Logic, Language and Computation, University of Amsterdam, Amsterdam, The Netherlands

    Dick de Jongh

  2. Department of Mathematics, Shahid Beheshti University, G.C., Evin, Tehran, Iran

    Fatemeh Shirmohammadzadeh Maleki

Authors
  1. Dick de Jongh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fatemeh Shirmohammadzadeh Maleki
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dick de Jongh .

Editor information

Editors and Affiliations

  1. Faculty of Technology, Policy and Management, Delft University of Technology, Delft, Zuid-Holland, The Netherlands

    Helle Hvid Hansen

  2. Department of Linguistics, Cornell University, Ithaca, New York, USA

    Sarah E. Murray

  3. School of Electronic Engineering and Computer Science, Queen Mary University of London, London, United Kingdom

    Mehrnoosh Sadrzadeh

  4. Heinrich-Heine-Universität, Düsseldorf, Germany

    Henk Zeevat

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer-Verlag GmbH Germany

About this paper

Cite this paper

de Jongh, D., Shirmohammadzadeh Maleki, F. (2017). Subintuitionistic Logics with Kripke Semantics. In: Hansen, H., Murray, S., Sadrzadeh, M., Zeevat, H. (eds) Logic, Language, and Computation. TbiLLC 2015. Lecture Notes in Computer Science(), vol 10148. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-54332-0_18

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-54332-0_18

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-54331-3

  • Online ISBN: 978-3-662-54332-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation